vii
TABLE OF CONTENTS
CHAPTER
1
TITLE
PAGE
DECLARATION
ii
DEDICATION
iii
ACKNOWLEDGEMENTS
iv
ABSTRACT
v
ABSTRAK
vi
TABLE OF CONTENTS
vii
LIST OF FIGURES
x
LIST OF TABLES
xi
LIST OF SYMBOLS
xii
INTRODUCTION
1
1.1 Introduction
1
1.2 Research Background
2
1.3 Problem Statement
3
1.4 Research Objectives
3
1.5 Scope of the Study
3
1.6 Significance of Findings
3
1.7 Research Methodology
4
1.8 Groups, Algorithms and Programming (GAP)
4
1.9 Thesis Organization
5
viii
2
3
4
5
LITERATURE REVIEW
7
2.1 Introduction
7
2.2 Some Basic Results
8
2.3 Conclusion
22
CAPABILITY OF INFINITE TWO-GENERATOR
GROUPS OF NILPOTENCY CLASS TWO
23
3.1 Introduction
23
3.2 Preliminaries
25
3.3 Capability of Infinite Two-Generator Groups of
Nilpotency Class Two
3.4 Conclusion
26
28
HOMOLOGICAL FUNCTORS OF INFINITE
TWO-GENERATOR GROUPS OF NILPOTENCY
CLASS TWO
29
4.1 Introduction
29
4.2 Preliminaries
29
4.3 Homological Functors of Infinite Two-Generator
Groups of Nilpotency Class Two
33
4.4 Composition Theorems
71
4.5 Conclusion
76
GAP PROGRAMMES
77
5.1 Introduction
77
5.2 Constructing Examples of Type 2.33 Groups
77
5.3 Constructing Examples of Type 2.34 Groups
81
5.4 Constructing Examples of Type 2.35 Groups
83
5.5 Constructing Examples of Type 2.36 Groups
86
5.6 Conclusion
88
ix
6
SUMMARY AND CONCLUSION
89
6.1 Summary of the Research
89
6.2 Suggestions for Future Research
90
REFERENCES
91
Appendix A GAP Simulation
94
Appendix B Publication/Presentation In Seminars/Conferences
171
x
LIST OF FIGURES
FIGURE NO.
2.1
TITLE
The Commutative Diagrams of Homological Functors
PAGE
22
xi
LIST OF TABLES
TABLE NO.
5.1
5.2
5.3
5.4
TITLE
PAGE
Different Values of α, γ and p in Theorem 4.6
to Theorem 4.8
81
Different Values of α, γ and p in Theorem 4.9
and Theorem 4.10
83
Different Values of α, γ, σ and p in Theorem 4.11
to Theorem 4.19
85
Different Values of α and p in Theorem 4.20
and Theorem 4.21
87
xii
LIST OF SYMBOLS
a|b
- a divides b
|G|, |x|
- Order of the group G, the order of the element x
Z
- Integers, the infinite cyclic group
Zn
- Cyclic group of order n
Z/nZ
- Integers modulo n
H≤G
- H is a subgroup of G
G∼
=H
- G is isomorphic to H
G⊕H
- Direct sum of G and H
G×H
- Direct product of G and H
GH
- Semidirect product of G and H
G⊗H
- Tensor product of G and H
G⊗G
- Tensor square of G
G ⊗Z G - Abelian tensor square of G
˜ G
G⊗
- Symmetric square of G
G∧G
- Exterior square of G
M(G)
- Schur multiplier of G
CG (A)
- Centralizer of A in G
Z(G)
- Center of the group G
x
- Group generated by the element x
ker(κ)
- Kernel of the homomorphism κ
N G
- N is a normal subgroup of G
gH, Hg
- Left coset, and right coset of H with coset representative g
|G : H|
- Index of the subgroup H in the group G
xiii
Inn(G)
- Inner automorphism group of G
T (G)
- Torsion subgroup of G
g
- The conjugate of h by g
h
[g, h]
- The commutator of g and h
G
- The commutator subgroup of G
⊂
- Proper subset
⊆
- Subset
∈
- Element of
∈
/
- Not element of
∩
- Intersection
∧
- Wedge product
1∧
- Identity of exterior square
=
- Not equal to
<
- Less than
≤
- Less than or equal to
>
- Greater than
≥
- Greater than or equal to
X
- Direct product